3.9.35 \(\int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx\)

Optimal. Leaf size=367 \[ \frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}+\frac {\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}}-\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

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Rubi [A]  time = 0.51, antiderivative size = 367, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {832, 779, 612, 621, 206} \begin {gather*} \frac {\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}-\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac {\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}}+\frac {x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac {x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

-((33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*(b + 2*c*x)*Sqrt[a
+ b*x + c*x^2])/(1024*c^6) + ((33*b^2*B - 42*A*b*c - 32*a*B*c)*x^2*(a + b*x + c*x^2)^(3/2))/(280*c^3) - ((11*
b*B - 14*A*c)*x^3*(a + b*x + c*x^2)^(3/2))/(84*c^2) + (B*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) + ((1155*b^4*B - 1
470*A*b^3*c - 3276*a*b^2*B*c + 2744*a*A*b*c^2 + 1024*a^2*B*c^2 - 6*c*(231*b^3*B - 294*A*b^2*c - 444*a*b*B*c +
280*a*A*c^2)*x)*(a + b*x + c*x^2)^(3/2))/(13440*c^5) + ((b^2 - 4*a*c)*(33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c +
112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(204
8*c^(13/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int x^4 (A+B x) \sqrt {a+b x+c x^2} \, dx &=\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\int x^3 \left (-4 a B-\frac {1}{2} (11 b B-14 A c) x\right ) \sqrt {a+b x+c x^2} \, dx}{7 c}\\ &=-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\int x^2 \left (\frac {3}{2} a (11 b B-14 A c)+\frac {3}{4} \left (33 b^2 B-42 A b c-32 a B c\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{42 c^2}\\ &=\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\int x \left (-\frac {3}{2} a \left (33 b^2 B-42 A b c-32 a B c\right )-\frac {3}{8} \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{210 c^3}\\ &=\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^5}\\ &=-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac {\left (\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^6}\\ &=-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac {\left (\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{1024 c^6}\\ &=-\frac {\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^6}+\frac {\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac {(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac {B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac {\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{13/2}}\\ \end {align*}

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Mathematica [A]  time = 0.54, size = 312, normalized size = 0.85 \begin {gather*} \frac {\frac {7 \left (32 a^2 A c^3-80 a^2 b B c^2-112 a A b^2 c^2+120 a b^3 B c+42 A b^4 c-33 b^5 B\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2048 c^{11/2}}+\frac {x^2 (a+x (b+c x))^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{40 c^2}+\frac {(a+x (b+c x))^{3/2} \left (252 b^2 c (7 A c x-13 a B)+8 a b c^2 (343 A+333 B x)+16 a c^2 (64 a B-105 A c x)-42 b^3 c (35 A+33 B x)+1155 b^4 B\right )}{1920 c^4}+\frac {x^3 (a+x (b+c x))^{3/2} (14 A c-11 b B)}{12 c}+B x^4 (a+x (b+c x))^{3/2}}{7 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

(((33*b^2*B - 42*A*b*c - 32*a*B*c)*x^2*(a + x*(b + c*x))^(3/2))/(40*c^2) + ((-11*b*B + 14*A*c)*x^3*(a + x*(b +
c*x))^(3/2))/(12*c) + B*x^4*(a + x*(b + c*x))^(3/2) + ((a + x*(b + c*x))^(3/2)*(1155*b^4*B - 42*b^3*c*(35*A +
33*B*x) + 8*a*b*c^2*(343*A + 333*B*x) + 16*a*c^2*(64*a*B - 105*A*c*x) + 252*b^2*c*(-13*a*B + 7*A*c*x)))/(1920
*c^4) + (7*(-33*b^5*B + 42*A*b^4*c + 120*a*b^3*B*c - 112*a*A*b^2*c^2 - 80*a^2*b*B*c^2 + 32*a^2*A*c^3)*(2*Sqrt[
c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/
(2048*c^(11/2)))/(7*c)

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IntegrateAlgebraic [A]  time = 1.82, size = 423, normalized size = 1.15 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (8192 a^3 B c^3+25312 a^2 A b c^3-6720 a^2 A c^4 x-34608 a^2 b^2 B c^2+12704 a^2 b B c^3 x-4096 a^2 B c^4 x^2-23520 a A b^3 c^2+12544 a A b^2 c^3 x-7616 a A b c^4 x^2+4480 a A c^5 x^3+21840 a b^4 B c-12096 a b^3 B c^2 x+7776 a b^2 B c^3 x^2-5056 a b B c^4 x^3+3072 a B c^5 x^4+4410 A b^5 c-2940 A b^4 c^2 x+2352 A b^3 c^3 x^2-2016 A b^2 c^4 x^3+1792 A b c^5 x^4+17920 A c^6 x^5-3465 b^6 B+2310 b^5 B c x-1848 b^4 B c^2 x^2+1584 b^3 B c^3 x^3-1408 b^2 B c^4 x^4+1280 b B c^5 x^5+15360 B c^6 x^6\right )}{107520 c^6}+\frac {\left (-128 a^3 A c^4+320 a^3 b B c^3+480 a^2 A b^2 c^3-560 a^2 b^3 B c^2-280 a A b^4 c^2+252 a b^5 B c+42 A b^6 c-33 b^7 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2048 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4*(A + B*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

(Sqrt[a + b*x + c*x^2]*(-3465*b^6*B + 4410*A*b^5*c + 21840*a*b^4*B*c - 23520*a*A*b^3*c^2 - 34608*a^2*b^2*B*c^2
+ 25312*a^2*A*b*c^3 + 8192*a^3*B*c^3 + 2310*b^5*B*c*x - 2940*A*b^4*c^2*x - 12096*a*b^3*B*c^2*x + 12544*a*A*b^
2*c^3*x + 12704*a^2*b*B*c^3*x - 6720*a^2*A*c^4*x - 1848*b^4*B*c^2*x^2 + 2352*A*b^3*c^3*x^2 + 7776*a*b^2*B*c^3*
x^2 - 7616*a*A*b*c^4*x^2 - 4096*a^2*B*c^4*x^2 + 1584*b^3*B*c^3*x^3 - 2016*A*b^2*c^4*x^3 - 5056*a*b*B*c^4*x^3 +
4480*a*A*c^5*x^3 - 1408*b^2*B*c^4*x^4 + 1792*A*b*c^5*x^4 + 3072*a*B*c^5*x^4 + 1280*b*B*c^5*x^5 + 17920*A*c^6*
x^5 + 15360*B*c^6*x^6))/(107520*c^6) + ((-33*b^7*B + 42*A*b^6*c + 252*a*b^5*B*c - 280*a*A*b^4*c^2 - 560*a^2*b^
3*B*c^2 + 480*a^2*A*b^2*c^3 + 320*a^3*b*B*c^3 - 128*a^3*A*c^4)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]
])/(2048*c^(13/2))

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fricas [A]  time = 0.56, size = 843, normalized size = 2.30 \begin {gather*} \left [\frac {105 \, {\left (33 \, B b^{7} + 128 \, A a^{3} c^{4} - 160 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 280 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 42 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (15360 \, B c^{7} x^{6} - 3465 \, B b^{6} c + 1280 \, {\left (B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 32 \, {\left (256 \, B a^{3} + 791 \, A a^{2} b\right )} c^{4} - 128 \, {\left (11 \, B b^{2} c^{5} - 2 \, {\left (12 \, B a + 7 \, A b\right )} c^{6}\right )} x^{4} - 336 \, {\left (103 \, B a^{2} b^{2} + 70 \, A a b^{3}\right )} c^{3} + 16 \, {\left (99 \, B b^{3} c^{4} + 280 \, A a c^{6} - 2 \, {\left (158 \, B a b + 63 \, A b^{2}\right )} c^{5}\right )} x^{3} + 210 \, {\left (104 \, B a b^{4} + 21 \, A b^{5}\right )} c^{2} - 8 \, {\left (231 \, B b^{4} c^{3} + 8 \, {\left (64 \, B a^{2} + 119 \, A a b\right )} c^{5} - 6 \, {\left (162 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (1155 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 16 \, {\left (397 \, B a^{2} b + 392 \, A a b^{2}\right )} c^{4} - 42 \, {\left (144 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{430080 \, c^{7}}, -\frac {105 \, {\left (33 \, B b^{7} + 128 \, A a^{3} c^{4} - 160 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 280 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 42 \, {\left (6 \, B a b^{5} + A b^{6}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (15360 \, B c^{7} x^{6} - 3465 \, B b^{6} c + 1280 \, {\left (B b c^{6} + 14 \, A c^{7}\right )} x^{5} + 32 \, {\left (256 \, B a^{3} + 791 \, A a^{2} b\right )} c^{4} - 128 \, {\left (11 \, B b^{2} c^{5} - 2 \, {\left (12 \, B a + 7 \, A b\right )} c^{6}\right )} x^{4} - 336 \, {\left (103 \, B a^{2} b^{2} + 70 \, A a b^{3}\right )} c^{3} + 16 \, {\left (99 \, B b^{3} c^{4} + 280 \, A a c^{6} - 2 \, {\left (158 \, B a b + 63 \, A b^{2}\right )} c^{5}\right )} x^{3} + 210 \, {\left (104 \, B a b^{4} + 21 \, A b^{5}\right )} c^{2} - 8 \, {\left (231 \, B b^{4} c^{3} + 8 \, {\left (64 \, B a^{2} + 119 \, A a b\right )} c^{5} - 6 \, {\left (162 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} + 2 \, {\left (1155 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 16 \, {\left (397 \, B a^{2} b + 392 \, A a b^{2}\right )} c^{4} - 42 \, {\left (144 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{215040 \, c^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/430080*(105*(33*B*b^7 + 128*A*a^3*c^4 - 160*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 280*(2*B*a^2*b^3 + A*a*b^4)*c^2
- 42*(6*B*a*b^5 + A*b^6)*c)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt
(c) - 4*a*c) + 4*(15360*B*c^7*x^6 - 3465*B*b^6*c + 1280*(B*b*c^6 + 14*A*c^7)*x^5 + 32*(256*B*a^3 + 791*A*a^2*b
)*c^4 - 128*(11*B*b^2*c^5 - 2*(12*B*a + 7*A*b)*c^6)*x^4 - 336*(103*B*a^2*b^2 + 70*A*a*b^3)*c^3 + 16*(99*B*b^3*
c^4 + 280*A*a*c^6 - 2*(158*B*a*b + 63*A*b^2)*c^5)*x^3 + 210*(104*B*a*b^4 + 21*A*b^5)*c^2 - 8*(231*B*b^4*c^3 +
8*(64*B*a^2 + 119*A*a*b)*c^5 - 6*(162*B*a*b^2 + 49*A*b^3)*c^4)*x^2 + 2*(1155*B*b^5*c^2 - 3360*A*a^2*c^5 + 16*(
397*B*a^2*b + 392*A*a*b^2)*c^4 - 42*(144*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, -1/215040*(10
5*(33*B*b^7 + 128*A*a^3*c^4 - 160*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 280*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 42*(6*B*a*
b^5 + A*b^6)*c)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(1
5360*B*c^7*x^6 - 3465*B*b^6*c + 1280*(B*b*c^6 + 14*A*c^7)*x^5 + 32*(256*B*a^3 + 791*A*a^2*b)*c^4 - 128*(11*B*b
^2*c^5 - 2*(12*B*a + 7*A*b)*c^6)*x^4 - 336*(103*B*a^2*b^2 + 70*A*a*b^3)*c^3 + 16*(99*B*b^3*c^4 + 280*A*a*c^6 -
2*(158*B*a*b + 63*A*b^2)*c^5)*x^3 + 210*(104*B*a*b^4 + 21*A*b^5)*c^2 - 8*(231*B*b^4*c^3 + 8*(64*B*a^2 + 119*A
*a*b)*c^5 - 6*(162*B*a*b^2 + 49*A*b^3)*c^4)*x^2 + 2*(1155*B*b^5*c^2 - 3360*A*a^2*c^5 + 16*(397*B*a^2*b + 392*A
*a*b^2)*c^4 - 42*(144*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^7]

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giac [A]  time = 0.24, size = 414, normalized size = 1.13 \begin {gather*} \frac {1}{107520} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B x + \frac {B b c^{5} + 14 \, A c^{6}}{c^{6}}\right )} x - \frac {11 \, B b^{2} c^{4} - 24 \, B a c^{5} - 14 \, A b c^{5}}{c^{6}}\right )} x + \frac {99 \, B b^{3} c^{3} - 316 \, B a b c^{4} - 126 \, A b^{2} c^{4} + 280 \, A a c^{5}}{c^{6}}\right )} x - \frac {231 \, B b^{4} c^{2} - 972 \, B a b^{2} c^{3} - 294 \, A b^{3} c^{3} + 512 \, B a^{2} c^{4} + 952 \, A a b c^{4}}{c^{6}}\right )} x + \frac {1155 \, B b^{5} c - 6048 \, B a b^{3} c^{2} - 1470 \, A b^{4} c^{2} + 6352 \, B a^{2} b c^{3} + 6272 \, A a b^{2} c^{3} - 3360 \, A a^{2} c^{4}}{c^{6}}\right )} x - \frac {3465 \, B b^{6} - 21840 \, B a b^{4} c - 4410 \, A b^{5} c + 34608 \, B a^{2} b^{2} c^{2} + 23520 \, A a b^{3} c^{2} - 8192 \, B a^{3} c^{3} - 25312 \, A a^{2} b c^{3}}{c^{6}}\right )} - \frac {{\left (33 \, B b^{7} - 252 \, B a b^{5} c - 42 \, A b^{6} c + 560 \, B a^{2} b^{3} c^{2} + 280 \, A a b^{4} c^{2} - 320 \, B a^{3} b c^{3} - 480 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*B*x + (B*b*c^5 + 14*A*c^6)/c^6)*x - (11*B*b^2*c^4 - 24*B*a*
c^5 - 14*A*b*c^5)/c^6)*x + (99*B*b^3*c^3 - 316*B*a*b*c^4 - 126*A*b^2*c^4 + 280*A*a*c^5)/c^6)*x - (231*B*b^4*c^
2 - 972*B*a*b^2*c^3 - 294*A*b^3*c^3 + 512*B*a^2*c^4 + 952*A*a*b*c^4)/c^6)*x + (1155*B*b^5*c - 6048*B*a*b^3*c^2
- 1470*A*b^4*c^2 + 6352*B*a^2*b*c^3 + 6272*A*a*b^2*c^3 - 3360*A*a^2*c^4)/c^6)*x - (3465*B*b^6 - 21840*B*a*b^4
*c - 4410*A*b^5*c + 34608*B*a^2*b^2*c^2 + 23520*A*a*b^3*c^2 - 8192*B*a^3*c^3 - 25312*A*a^2*b*c^3)/c^6) - 1/204
8*(33*B*b^7 - 252*B*a*b^5*c - 42*A*b^6*c + 560*B*a^2*b^3*c^2 + 280*A*a*b^4*c^2 - 320*B*a^3*b*c^3 - 480*A*a^2*b
^2*c^3 + 128*A*a^3*c^4)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)

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maple [B]  time = 0.06, size = 872, normalized size = 2.38 \begin {gather*} \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,x^{4}}{7 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,x^{3}}{6 c}-\frac {11 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b \,x^{3}}{84 c^{2}}+\frac {A \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {15 A \,a^{2} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {7}{2}}}+\frac {35 A a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}-\frac {21 A \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {11}{2}}}-\frac {5 B \,a^{3} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {7}{2}}}+\frac {35 B \,a^{2} b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}-\frac {63 B a \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {11}{2}}}+\frac {33 B \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {13}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} x}{16 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{2} x}{32 c^{3}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4} x}{256 c^{4}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b \,x^{2}}{20 c^{2}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b x}{32 c^{3}}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{3} x}{64 c^{4}}-\frac {4 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a \,x^{2}}{35 c^{2}}-\frac {33 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5} x}{512 c^{5}}+\frac {33 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} x^{2}}{280 c^{3}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} b}{32 c^{3}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{3}}{64 c^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a x}{8 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{5}}{512 c^{5}}+\frac {21 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} x}{160 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b^{2}}{64 c^{4}}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{4}}{128 c^{5}}+\frac {111 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b x}{560 c^{3}}-\frac {33 \sqrt {c \,x^{2}+b x +a}\, B \,b^{6}}{1024 c^{6}}-\frac {33 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3} x}{320 c^{4}}+\frac {49 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a b}{240 c^{3}}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{64 c^{4}}+\frac {8 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,a^{2}}{105 c^{3}}-\frac {39 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a \,b^{2}}{160 c^{4}}+\frac {11 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{4}}{128 c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x)

[Out]

-4/35*B*a/c^2*x^2*(c*x^2+b*x+a)^(3/2)-11/84*B*b/c^2*x^3*(c*x^2+b*x+a)^(3/2)-5/32*B*b/c^3*a^2*(c*x^2+b*x+a)^(1/
2)*x+15/64*B*b^3/c^4*a*(c*x^2+b*x+a)^(1/2)*x-7/32*A*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*x+111/560*B*b/c^3*a*x*(c*x^2
+b*x+a)^(3/2)+1/16*A*a^2/c^2*(c*x^2+b*x+a)^(1/2)*x-1/8*A*a/c^2*x*(c*x^2+b*x+a)^(3/2)+1/32*A*a^2/c^3*(c*x^2+b*x
+a)^(1/2)*b+21/256*A*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x+15/128*B*b^4/c^5*a*(c*x^2+b*x+a)^(1/2)-15/64*A*b^2/c^(7/2)*
a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+35/128*B*b^3/c^(9/2)*a^2*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))-5/32*B*b/c^(7/2)*a^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-39/160*B*b^2/c^4*a*(c*x^2+b*x+a)^(3/2)
+33/280*B*b^2/c^3*x^2*(c*x^2+b*x+a)^(3/2)-33/320*B*b^3/c^4*x*(c*x^2+b*x+a)^(3/2)-33/512*B*b^5/c^5*(c*x^2+b*x+a
)^(1/2)*x+21/160*A*b^2/c^3*x*(c*x^2+b*x+a)^(3/2)-5/64*B*b^2/c^4*a^2*(c*x^2+b*x+a)^(1/2)-63/512*B*b^5/c^(11/2)*
ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+35/256*A*b^4/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*a-7/64*A*b^3/c^4*a*(c*x^2+b*x+a)^(1/2)+49/240*A*b/c^3*a*(c*x^2+b*x+a)^(3/2)-3/20*A*b/c^2*x^2*(c*x^2+b*x+a)^(3
/2)+1/6*A*x^3*(c*x^2+b*x+a)^(3/2)/c-7/64*A*b^3/c^4*(c*x^2+b*x+a)^(3/2)+8/105*B*a^2/c^3*(c*x^2+b*x+a)^(3/2)+33/
2048*B*b^7/c^(13/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+11/128*B*b^4/c^5*(c*x^2+b*x+a)^(3/2)-33/1024*B
*b^6/c^6*(c*x^2+b*x+a)^(1/2)+21/512*A*b^5/c^5*(c*x^2+b*x+a)^(1/2)-21/1024*A*b^6/c^(11/2)*ln((c*x+1/2*b)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))+1/16*A*a^3/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/7*B*x^4*(c*x^2+b*x+a)^
(3/2)/c

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [B]  time = 3.91, size = 992, normalized size = 2.70 \begin {gather*} \frac {8\,B\,a^3\,\sqrt {c\,x^2+b\,x+a}}{105\,c^3}-\frac {33\,B\,b^6\,\sqrt {c\,x^2+b\,x+a}}{1024\,c^6}+\frac {A\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{6\,c}+\frac {B\,x^4\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{7\,c}+\frac {33\,B\,b^7\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{2048\,c^{13/2}}+\frac {A\,a\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{2\,c}-\frac {3\,A\,b\,\left (\frac {7\,b\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}-\frac {2\,a\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}+\frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}\right )}{4\,c}-\frac {5\,B\,a^3\,b\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{32\,c^{7/2}}-\frac {63\,B\,a\,b^5\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{512\,c^{11/2}}+\frac {35\,B\,a^2\,b^3\,\ln \left (b+2\,\sqrt {c}\,\sqrt {c\,x^2+b\,x+a}+2\,c\,x\right )}{128\,c^{9/2}}+\frac {13\,B\,a\,b^4\,\sqrt {c\,x^2+b\,x+a}}{64\,c^5}-\frac {4\,B\,a\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{35\,c^2}-\frac {11\,B\,b\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{84\,c^2}-\frac {33\,B\,b^3\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{320\,c^4}+\frac {11\,B\,b^5\,x\,\sqrt {c\,x^2+b\,x+a}}{512\,c^5}-\frac {103\,B\,a^2\,b^2\,\sqrt {c\,x^2+b\,x+a}}{320\,c^4}+\frac {8\,B\,a^2\,x^2\,\sqrt {c\,x^2+b\,x+a}}{105\,c^2}+\frac {33\,B\,b^2\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{280\,c^3}+\frac {11\,B\,b^4\,x^2\,\sqrt {c\,x^2+b\,x+a}}{128\,c^4}-\frac {39\,B\,a\,b^2\,x^2\,\sqrt {c\,x^2+b\,x+a}}{160\,c^3}+\frac {111\,B\,a\,b\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{560\,c^3}-\frac {269\,B\,a^2\,b\,x\,\sqrt {c\,x^2+b\,x+a}}{3360\,c^3}-\frac {3\,B\,a\,b^3\,x\,\sqrt {c\,x^2+b\,x+a}}{320\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(A + B*x)*(a + b*x + c*x^2)^(1/2),x)

[Out]

(8*B*a^3*(a + b*x + c*x^2)^(1/2))/(105*c^3) - (33*B*b^6*(a + b*x + c*x^2)^(1/2))/(1024*c^6) + (A*x^3*(a + b*x
+ c*x^2)^(3/2))/(6*c) + (B*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) + (33*B*b^7*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^
(1/2) + 2*c*x))/(2048*c^(13/2)) + (A*a*((5*b*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a
*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(8*c) - (x*(a +
b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x
+ c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(2*c) - (3*A*b*((7*b*((5*b*((log((b + 2*c*x)/c^(1/2) + 2
*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^
2)^(1/2))/(24*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) +
(log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(10*c) - (2*a*((log(
(b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2
*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) + (x^2*(a + b*x + c*x^2)^(3/2))/(5*c)))/(4*c) - (5*B*a^3*b*l
og(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x))/(32*c^(7/2)) - (63*B*a*b^5*log(b + 2*c^(1/2)*(a + b*x + c*x
^2)^(1/2) + 2*c*x))/(512*c^(11/2)) + (35*B*a^2*b^3*log(b + 2*c^(1/2)*(a + b*x + c*x^2)^(1/2) + 2*c*x))/(128*c^
(9/2)) + (13*B*a*b^4*(a + b*x + c*x^2)^(1/2))/(64*c^5) - (4*B*a*x^2*(a + b*x + c*x^2)^(3/2))/(35*c^2) - (11*B*
b*x^3*(a + b*x + c*x^2)^(3/2))/(84*c^2) - (33*B*b^3*x*(a + b*x + c*x^2)^(3/2))/(320*c^4) + (11*B*b^5*x*(a + b*
x + c*x^2)^(1/2))/(512*c^5) - (103*B*a^2*b^2*(a + b*x + c*x^2)^(1/2))/(320*c^4) + (8*B*a^2*x^2*(a + b*x + c*x^
2)^(1/2))/(105*c^2) + (33*B*b^2*x^2*(a + b*x + c*x^2)^(3/2))/(280*c^3) + (11*B*b^4*x^2*(a + b*x + c*x^2)^(1/2)
)/(128*c^4) - (39*B*a*b^2*x^2*(a + b*x + c*x^2)^(1/2))/(160*c^3) + (111*B*a*b*x*(a + b*x + c*x^2)^(3/2))/(560*
c^3) - (269*B*a^2*b*x*(a + b*x + c*x^2)^(1/2))/(3360*c^3) - (3*B*a*b^3*x*(a + b*x + c*x^2)^(1/2))/(320*c^4)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \left (A + B x\right ) \sqrt {a + b x + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(x**4*(A + B*x)*sqrt(a + b*x + c*x**2), x)

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